Measurability of optimal transportation and convergence rate for Landau type interacting particle systems

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Measurability of optimal transportation and convergence rate for Landau type interacting particle systems

Joaquin Fontbona, Hélène Guérin, Sylvie Méléard

To cite this version:

Joaquin Fontbona, Hélène Guérin, Sylvie Méléard. Measurability of optimal transportation and con- vergence rate for Landau type interacting particle systems. Probability Theory and Related Fields, Springer Verlag, 2009, 143 (3-4), pp.329-351. �10.1007/s00440-007-0128-4�. �hal-00139882�

Measurability of optimal transportation and convergence rate for Landau type interacting

particle systems

Joaquin Fontbona^∗, H´el`ene Gu´erin ^†, Sylvie M´el´eard^‡

Abstract

In this paper, we consider nonlinear diffusion processes driven by space-time white noises, which have an interpretation in terms of partial differential equations. For a specific choice of coefficients, they correspond to the Landau equation arising in kinetic theory. A particular feature is that the diffusion matrix of this process is a linear function the law of the process, and not a quadratic one, as in the McKean- Vlasov model. The main goal of the paper is to construct an easily simulable diffusive interacting particle system, converging towards this nonlinear process and to obtain an explicit pathwise rate. This requires to find a significant coupling between finitely many Brownian motions and the infinite dimensional white noise process. The key idea will be to construct the right Brownian motions by pushing forward the white noise processes, through the Brenier map realizing the optimal transport between the law of the nonlinear process, and the empirical measure of independent copies of it. A striking problem then is to establish the joint measurability of this optimal transport map with respect to the space variable and the parameters (time and randomness) making the marginals vary. We shall prove a general measurability result for the mass transportation problem in terms of the support of the transfert plans, in the sense of set-valued mappings. This will allow us to construct the coupling and to obtain explicit convergence rates.

Key words and phrases: Landau type interacting particle systems, nonlinear white noise driven SDE, pathwise coupling, measurability of optimal transport, predictable transport process.

MSC: 60K35, 49Q20, 82C40, 82C80, 60G07.

1 Introduction and main statements

Consider the nonlinear diffusion processes inR^d of the following type:

X_t=X₀+ Z t

0

Z

R^d

σ(X_s−y)W_P(dy, ds) + Z t

0

Z

R^d

b(X_s−y)P_s(dy)ds (1)

∗DIM-CMM, Universidad de Chile, Casilla 170-3, Correo 3, Santiago-Chile, e- mail:fontbona@dim.uchile.cl. Supported by Fondecyt Proyect 1040689, ECOS-Conicyt C05E02, Millennium Nucleus Information and Randomness ICM P04-069-F and FONDAP Applied Mathematics

†IRMAR, Universit´e Rennes 1, Campus de Beaulieu, 35042 Rennes-France, e-mail:helene.guerin@univ- rennes1.fr. Supported by ECOS-Conicyt C05E02 and Millennium Nucleus Information and Randomness ICM P04-069-F

‡CMAP, Ecole Polytechnique, CNRS, route de Saclay, 91128 Palaiseau Cedex-France e-mail:

sylvie.meleard@polytechnique.edu. Supported by ECOS-Conicyt C05E02 and Millennium Nucleus Infor- mation and Randomness ICM P04-069-F

where Pt is the law of Xt, and WP is a R^d valued space-time white noise on [0, T]×R^d with independent coordinates, each of which having covariance measureP_t(dy)⊗dt.

The nonlinear process (1) was introduced by Funaki [3], who obtained existence and unique- ness results for Lipschitz coefficients σ :R^d→ R^d⊗d and b :R^d →R^d, see also Guerin [7]

for a different approach. It has an important interpretation in terms of partial differential equations issued from kinetic theory. More precisely, for a specific choice of coefficientsσ andb, the laws (P_t)_tare a weak solution of the spatially homogeneous Landau (also called Fokker-Planck-Landau) equations for Maxwell potential:

∂f

∂t (t, v) = 1 2

d

X

i,j=1

∂

∂v_i Z

R^d

a_ij(v−v∗)

f(t, v∗) ∂f

∂v_j (t, v)−f(t, v) ∂f

∂v∗j

(t, v∗)

dv∗

, (2) with aij(v) := (σσ^∗)ij(v) =|v|²δij −vivj and bi(v) =∇ ·ai·(v). The equations (2) model collisions of particles in a plasma and can be obtained as limit of the Boltzmann equations when collisions become grazing, see Funaki [4], Goudon [5], Villani [17] [18] and Gu´erin- M´el´eard [8].

In this work, we shall prove the convergence in law of an easily simulable mean field inter- acting particle system towards the nonlinear process (1) at an explicit pathwise rate. This problem is of great interest in order to construct a tractable simulation algorithm for the law P_t and thus, in particular, for solutions f of equation (2). To our knowledge, there is no result on convergence rates of the deterministic numerical methods used at present for the Landau equation, which are reviewed in [2]. The interest of our approach is that it is based on the diffusive nature of the equation, and that it addresses a large class of nonlinear processes. The fact that we want to deal with simulable systems will necessitate a coupling between finite dimensional and infinite dimensional stochastic processes. We shall introduce a coupling argument based on new results on measurability of the optimal mass transportation problem.

We consider a particle system which is naturally related to the nonlinear process. Indeed, notice that the diffusion matrix associated with (1) is defined onR^dby

a(x, P_t) :=

Z

R^d

σ(x−y)σ^∗(x−y)P_t(dy) = [(σσ^∗)∗P_t](x). (3) Thus, if in order to approximate the white noise driven stochastic differential equation (1), we heuristically replacePt in (3) by an empirical measure ofn∈N^∗ particles inR^d, we are led to consider the following system driven byn² independent Brownian motions (B^ik):

X_t^i,n=X₀ⁱ+ 1

√n Z t

0 n

X

k=1

σ(X_s^i,n−X_s^k,n)dB_s^ik+1 n

Z t 0

n

X

k=1

b(X_s^i,n−X_s^k,n)ds, i= 1, . . . , n. (4) To be more precise, if µⁿ_t = ¹_nPn

i=1δ_X^i,n

t is the empirical measure of the system, the mappings

f(t, ω, x)7→ 1

√n Z t

0 n

X

k=1

f(s, ω, X_s^k,n)dB_s^ik, i= 1, . . . , n, (5) define (for suitably measurable functions f) orthogonal martingale measures in the sense of Walsh [20], with covariance measureµⁿ_t ⊗dt.

By adapting techniques of M´el´eard-Roelly [11] based on martingale problems, one can show propagation of chaos for system (4) with as limit the process (1). This says in particular that the covariance measure of (5) converges in law toPt⊗dt whenngoes to infinity. But in turn, the arguments of [11] do not give any information about speed of convergence.

To estimate the distance between the law of the particles and the law of the nonlinear process, we need to construct a significant coupling between finitely many Brownian motions and the white noises processes. This problem is much more subtle than in the McKean- Vlasov model (cf. Sznitman [16] or M´el´eard [12]), where each particle is coupled with a limiting process through a single Brownian motion that drives them both. The well known

√1

n−convergence rate in that model is consequence of the standardL²-law of large numbers inR^dand of the fact that the diffusion and drift coefficients of the nonlinear process depend linearly on the limiting law through expectations with respect to it. In the present Landau model, we have to deal with the space-time random fields (5), which have fluctuations of constant order inn. This is also reflected in the fact that it is thesquared diffusion matrix of (1), that depends linearly onP_t(see (3)). It is hence not clear where a convergence rate can be deduced from.

Let Xⁱ,i = 1, . . . , n be n independent copies of the nonlinear process in some probability space, and ν_tⁿ their empirical measure at time t (observe that it samples P_t). We shall construct particles (4) on the same probability space, in such way that they will converge pathwise inL² on finite time intervals, at the same rate at which the Wasserstein distance W₂ betweenP_t and ν_tⁿ goes to 0. Let us state our main result on the process (1):

Theorem 1.1. Let n∈N and assume usual Lipschitz hypothesis on σ and b, and that the law P₀ of X₀ⁱ has finite second order moment. Assume moreover that P_t has a density with respect to Lebesgue measure for each t >0.

Then, in the same probability space as(X¹, . . . , Xⁿ)there exist independent standard Brow- nian motions (B^ik)1≤i,k≤n such that the particle system (X^i,n)ⁿ_i=1 defined in (4) satisfies

E sup

t∈[0,T]

|X_t^i,n−X_tⁱ|²

!

≤Cexp(C⁰T) Z T

0

E(W₂²(ν_sⁿ, Ps))ds for constants C, C⁰ that do not depend on n.

Thanks to available convergence results for empirical measures of i.i.d samples (see e.g.

[14]), Theorem 1.1 will allow us to obtain, under some additional moment assumptions on P0, the speed of convergence n^d+4−2 for the pathwise law of the system (see Corollary 6.2).

We remark that the absolute continuity condition of Theorem 1.1 can be obtained under non-degeneracy of the matrix σσ^∗ by using for instance Malliavin calculus [13]; it is also true for the specific coefficients of the Landau equation (2) despite their degeneracy, and for some generalizations (see Gu´erin [6]).

The proof of Theorem 1.1 relies on new results on the optimal mass transportation problem.

For general background on the theory of mass transportation, we refer to Villani [19].

Recall that ifµ and ν are probability measures in R^d with finite second moment, the first of them having a density, then the optimal mass transportation problem with quadratic cost betweenµ and ν has a unique solution, which is a probability measure onR^2d of the formπ(dx, dy) =µ(dx)δ_T(x)(dy) . The so-called Brenier or optimal transport mapT(x) is (µa.s. equal to) the gradient of some convex function in R^d, and pushes forward µtoν.

Let now W_Pⁱ be the white noise process driving the i-th nonlinear process Xⁱ. The key idea in Theorem 1.1 will be to construct Brownian motions (B^ik)_k=1...n in an “optimal”

pathwise way from W_Pⁱ. Heuristically, this will consist in pushing forward the martingale measure W_Pⁱ through the Brenier maps T^t,ω,n(x) realizing the optimal transport between P_t andν_tⁿ(ω) (this is the reason for the absolute continuity assumption onP_t). But to give such a construction a rigorous sense, we must make sure that we can compute stochastic integrals of T^t,ω,n(x) with respect to W_Pⁱ(dx, dt). From the basic definition of stochastic integration with respect to space-time white noise (cf. [20]), this requires the existence of a measurable version of (t, ω, x)7→T^t,ω,n(x) being moreover predictable in (t, ω). A striking problem then is that no available result in the mass transportation theory can provide any information about joint measurability properties of the optimal transport map, with respect to the space variable and some parameter making the marginals vary. Nevertheless, we will show that a suitable “predictable transportation process” exists:

Theorem 1.2. There exists a measurable process (t, ω, x)7→Tⁿ(t, ω, x) that is predictable in (t, ω) with respect to the filtration associated to (W_P¹, . . . , W_Pⁿ) and (X₀¹, . . . , X₀ⁿ), and such that for dt⊗P(dω) almost every(t, ω),

Tⁿ(t, ω, x) =T^t,ω,n(x) P_t(dx)-almost surely.

This statement is consequence of a general abstract result about “measurability” of the mass transportation problem. To be more explicit, recall that the optimality of a transfert planπ is determined by its support (it is equivalent to the support being cyclically monotone, see McCann [10] or Villani [19]). On the other hand, without assumptions (besides moments) on the marginals µ and ν, the solution π of the mass transportation problem may not be unique. A basic question then is how to formulate, in a general setting, the adequate property of “measurability” of the solution(s)π with respect to the data (µ, ν). As we shall see, the natural formulation requires to introduce notions and techniques from set-valued analysis. Then, we shall prove the following

Theorem 1.3. Let P₂(R^d) be the space of Borel probability measures in R² with finite second order moment, endowed with the Wasserstein distance and its Borelσ−field. Denote by Π^∗(µ, ν) the set of solutions of the mass transportation problem with quadratic cost associated with (µ, ν)∈(P₂(R^d))². The function assigning to (µ, ν) the set of R^2d:

[

π∈Π^∗(µ,ν)

supp(π),

is measurable in the sense of set-valued mappings.

In particular, this ensures that ifµ_λ andν_λ vary in a measurable way with respect to some parameterλ, so that in each of the associated optimal transportation problems uniqueness holds, then the support of the solutionπ_λ also “varies” in a measurable way. This will be the key to our results.

The rest of this work is organized as follows. In Section 2 we review the Wasserstein distance and the mass transportation problem with quadratic cost inR^d (in particular the characterization of its minimizers). In Section 3 we prove Theorem 1.3 and a consequence needed to prove Theorem 1.1. In Section 4, we state some properties about process (1) and we heuristically describe our coupling between space-time white noises and Brownian motions. In Section 5 we construct the “predictable transportation process” of Theorem 1.2 needed to rigorously define the coupling. Section 6 is devoted to complete the proof of Theorem 1.1 and to obtain explicit convergence rates.

2 The mass transportation problem with quadratic cost in R^d and the Wasserstein distance

We denote the space of Borel probability measures in R^d by P(R^d), and by P₂(R^d) the subspace of probability measures having finite second order moment.

Givenπ ∈ P₂(R^2d), we respectively denote by π1 and π2 its first and second marginals on R^d. On the other hand, for any two probability measuresµ, ν ∈ P₂(R^d) and π ∈ P₂(R^2d), we write

π <^µ_ν

ifπ₁ =µand π₂ =ν. Suchπ is refereed to as a “transfert plan” between µand ν.

Definition 2.1. The Wassertein distance W2 onP₂(R^d) is defined by W₂²(µ, ν) := inf

π<^µν

Z

R²

|x−y|²π(dx, dy).

Then, (P₂(R^d), W₂) is a Polish space, see e.g. Rachev and R¨uschendorf [14]. The topology is stronger that the usual weak topology. More precisely, one has the following result (see for instance Villani, [19] Theorem 7.12)

Theorem 2.2. Let µⁿ, µ∈ P(R^d). The following are then equivalent:

i) W2(µⁿ, µ)→0 when n→ ∞.

ii) µⁿ converges weakly to µ and Z

R^d

|x|²µⁿ(dx)→ Z

R^d

|x|²µ(dx).

iii) We have

Z

R^d

ϕ(x)µⁿ(dx)→ Z

R^d

ϕ(x)µ(dx)

for all continuous functionϕ:R^d→Rsuch that|ϕ(x)| ≤C(1+|x|²)for someC∈R. We shall denote by Lthe mapping L:P₂(R^2d)→Rdefined by

L(π) = Z

R²

|x−y|²π(dx, dy).

Remark 2.3. It is not hard to check that L is lower semi continuous (l.s.c) for the weak topology. Moreover, Lis continuous for the Wasserstein topology in P₂(R^2d) by part iii) of Theorem 2.2.

Fix nowµ, ν ∈ P₂(R^d), and denote by Π^∗(µ, ν) the subset of P₂(R^2d) of minimizers of the Monge-Kantorovich transportation problem with quadratic cost for the pair of marginals (µ, ν) . That is,

Π^∗(µ, ν) :=argmin_π<^µ_νL(π).

It is well known that Π^∗(µ, ν) is non-empty. Indeed, it is not hard to see that for the weak topology,{π∈ P₂(R^2d) :π <^µν}is a compact set, and the lower semi-continuity ofLimplies the existence of minimizers (see e.g. [19] Chapter 1 for details).

We shall next recall the characterization of minimizers of the transportation problem with quadratic cost. We need the notion of sub-differential of a convex function:

Definition 2.4. Let ϕ : A ⊂ R^d →]− ∞,∞] be a proper (i.e. ϕ 6≡ +∞) lower semi- continuous (l.s.c) convex function. The sub-differential of ϕat x is

∂ϕ(x) ={y ∈R^d:ϕ(z)≥ϕ(x) +hy, z−xi,∀z∈R^d}.

Elements of ∂ϕ(x) are called sub-gradients of ϕ at point x. The graph of ∂ϕ is Gr(∂ϕ) ={(x, y)∈R^2d:y ∈∂ϕ(x)}

and it is a closed set.

Recall thatϕis differentiable atxif and only if∂ϕ(x) is a singleton (in which case∂ϕ(x) = {∇ϕ(x)}). Also, the set {x ∈ R^d : ϕis differentiable atx} is borelian, see e.g. McCann [10].

We next summarize results in pioneer works in this domain, Knott-Smith [9], Brenier [1] and McCann [10], Rachev and R¨uschendorf [14]. See also Villani [19] for a complete discussion on these questions, proofs and background.

Theorem 2.5. Let µ, ν∈ P(R^d) and π <^µν be a transfert plan. We have

a) π∈Π^∗(µ, ν) if and only if there exists a proper l.s.c. convex functionϕ such that supp(π)⊂Gr(∂ϕ)

or, equivalently

π({(x, y)∈R²:y ∈∂ϕ(x)}) = 1.

b) Assume that µ does not charge sets of Hausdorff dimension less or equal than d−1 and that π∈Π^∗(µ, ν). Then,

i) the set{x∈R^d:ϕis not differentiable at x} has null µ-measure.

ii) We have

π(dx, dy) =µ(dx)⊗δ_∇ϕ(x)(dy).

ii) If T is a measurable mapping such that π(dx, dy) = µ(dx)⊗δ_T(x)(dy), then T(x) =∇ϕ(x) , µ(dx)−a.s..

iii) π∈Π^∗(µ, ν) is unique.

This result will be useful later in the particular case when the measure µ is absolutely continuous with respect to Lebesgue measure.

3 Measurability of the mass transportation problem

We now introduce the basic notions on “multi-applications” or “set-valued mappings” that we need to prove Theorem 1.3. For general background, we refer the reader to Appendix A in Rockafellar and Wets [15].

Definition 3.1. Let X, Y be two sets.

i) A function S on X taking values in the set of subsets of Y is called a set-valued mapping or multi-application. We write S:X ⇒Y.

ii) For any A⊂Y, the inverse image of A through S is the set S⁻¹(A) :={x∈X:S(x)∩A6=∅}.

iii) If(X,A)is a measurable space and (Y,Θ)a topological space, we say thatS :X⇒Y is measurable if for allθ∈Θ,

S⁻¹(θ)∈ A.

(Of course, ifS(x) ={s(x)}is singleton for all x, measurability of S is equivalent to that ofs. )

ConsiderP₂(R^d) endowed with the Wasserstein distance and the Borelσ−field. We define a set-valued mapping

Ψ : (P₂(R^d))² ⇒R^2d by

Ψ(µ, ν) :={(x, y) :∃π∈Π^∗(µ, ν)s.t. (x, y)∈supp(π)}.

Our goal is to prove that Ψ is measurable. We shall need some further notions on set-valued mappings.

Definition 3.2. Let X be a set, and (Y,Ξ) and (Z,Θ) be topological spaces.

i) A set-valued mapping S :X ⇒ Y is closed-valued if for all x ∈ X, S(x) is a closed set of(Y,Ξ).

ii) A set-valued mapping U :Y ⇒Z is inner semicontinous (i.s.c) if for all θ∈Θ, S⁻¹(θ)∈Ξ

The following results can be found in Appendix A of [15], in the case of set-valued mappings inR^d. For completeness we provide proofs in a more general context.

Lemma 3.3. Let (X,A) be a measurable space and (Y,Ξ) a topological space.

i) S : X ⇒ Y is measurable if and only if the closed-valued mapping x ⇒ Cl(S(x)) is measurable, whereCl(S(x)) is the topological closure of the set S(x).

ii) Assume that (Y, d) is a separable metric space and that S :X ⇒ Y is closed-valued.

Then,S is measurable if and only if for all closed set F of Y, S⁻¹(F)∈ A.

iii) Let (Y,Ξ) and(Z,Θ) be topological spaces, S:X ⇒Y be measurable and U :Y ⇒Z be i.s.c. Then, the multi-applicationU ◦S:X ⇒Z, defined by

U ◦S(x) := [

y∈S(x)

U(y) is measurable.

Proofi)For any open setθ∈Ξ,S(x)∩θ6=∅if and only if Cl(S(x))∩θ6=∅.

ii)“Only if” part: sinceY is a metric space, we use that every closed setF is the intersection of some countable collection of open sets (θn). Therefore,

{x∈X:S(x)∩F 6=∅}= \

n∈N

{x∈X:S(x)∩θn6=∅} ∈ A.

“If” part: (Y, d) being separable, we can express every open set θ as the union of some countable collection (Bn) of closed balls. We then have that

{x∈X:S(x)∩θ6=∅}= [

n∈N

{x∈X :S(x)∩Bn6=∅} ∈ A.

iii)Straightforward:

(U ◦S)⁻¹(θ) = {x∈X: ∪_y∈S(x)U(y)

∩θ6=∅}={x∈X: ∃y∈S(x) s.t. U(y)∩θ6=∅}

= {x∈X:S(x)∩(U⁻¹(θ))6=∅}.

The functionU being i.s.c.,U⁻¹(θ) belongs to Ξ, which allows us to conclude.

Now we can proceed to the Proof of Theorem 1.3

We observe first that Ψ(µ, ν) =U ◦S(µ, ν), where S and U are the set valued mappings respectively defined by

(µ, ν)⇒S(µ, ν) := Π^∗(µ, ν) and U :P₂(R^2d)⇒R^d by

U(π) :=supp(π) We will therefore split the proof in several parts:

a) S is a closed valued mapping

First notice thatπ7→π_i is continuous for the Wasserstein topology. Indeed,W₂(πⁿ, π)→0 implies thatπⁿconverges weakly toπ, and thenπⁿ_i converges weakly toπ_ifori= 1,2. More- over, we haveR

R^d|x|²π₁ⁿ(dx) =R

R^2d|x|²πⁿ(dx, dy)→R

R^2d|x|²π(dx, dy) =R

R^d|x|²π1(dx) by Theorem 2.2, and then the asserted continuity follows.

Consequently,π 7→W₂(π₁, π₂) too is continuous. Therefore,

Π^∗(µ, ν) ={π :π <^µ_ν} ∩ {π :L(π)−W₂(π₁, π₂) = 0}

is the intersection of two closed setsP₂(R^2d).

b) Inverse images throughS of closed sets are closed sets

LetF ⊂ P₂(R^d) be a closed set and (µⁿ, νⁿ) ∈S⁻¹(F), n∈ N, be a sequence converging to (µ, ν) in (P₂(R^d))². Then, µⁿ→µ andνⁿ→ν weakly, and (µⁿ) and (νⁿ) are tight.

But since (µⁿ, νⁿ) ∈S⁻¹(F) for each n, there existsπ_n s.t. πⁿ<^µ_νnⁿ, and then (π_n) too is tight (by considering products of compact sets).

Let (π^nk) be a weakly convergent subsequence with limit π. Then, clearly π <^µν. We will prove that L(π) = W₂(µ, ν) and that π ∈ F, which will mean that (µ, ν) ∈ S⁻¹(F) and finish the proof.

We have Z

R^2d

|x|²+|y|²

π^nk(dx, dy) = Z

R^d

|x|²µ^nk(dx) + Z

R^d

|y|²ν^nk(dy)→ Z

R^d

|x|²µ(dx) + Z

R^d

|y|²ν(dy) = Z

R^2d

|x|²+|y|²

π(dx, dy),

which implies that W₂(πⁿ, π) → 0 and π ∈ F. Finally, by the continuity of π 7→ L(π)− W2(π1, π2) we get that

0 =L(π^nk)−W₂(π₁^nk, π₂^nk) =L(π)−W₂(µ, ν).

c) The mappingU is i.s.c.

Letθ be an open set ofR^2d. We must check that

{π∈ P₂(R^2d) :supp(π)∩θ6=∅}={π ∈ P₂(R^2d) :π(θ)>0}

is open, or equivalently, that

{π ∈ P₂(R^2d) :π(θ) = 0}

is closed in P₂(R^2d). Assume that π, πⁿ ∈ P₂(R^2d), with πⁿ such that πⁿ(θ) = 0 for all n∈N, and moreover thatW₂(πⁿ, π) → 0. Then πⁿ converges weakly toπ, and so by the Portemanteau theorem, we have

0 = lim inf

n πⁿ(θ)≥π(θ).

d) Conclusion

By partsa) and b)and Lemma 3.3 ii)we get thatS is measurable. Byc)and Lemma 3.3 iii)U ◦S is measurable and the proof is finished.

The following corollary will be useful in the specific setting needed to prove Theorem 1.1:

Corollary 3.4. Let (E,Σ) be a measurable space, and λ∈E 7→(µ_λ, ν_λ)∈(P₂(R^d))² and ξ:E →R^d be measurable functions. Then, the set

{(λ, x) : (x, ξ(λ))∈Cl(Ψ)(µ_λ, ν_λ)}

belongs toΣ⊗ B(R^d)

ProofBy Lemma 3.3 i) and Theorem 1.3 we get that Cl(Ψ) is measurable. Moreover, it is not hard to check that the mapping

(λ, x)⇒Cl(Ψ)(µ_λ, ν_λ)−(x, ξ(λ)) is measurable and closed-valued. Then, we just have to notice that

(x, ξ(λ))∈Cl(Ψ)(µ_λ, ν_λ) if and only if [Cl(Ψ)(µ_λ, ν_λ)−(x, ξ(λ))]∩C6=∅ for the closed setC={0}.

4 A coupling between space-time white noise and Brownian motions via optimal transport

In all the sequel, we refer the reader to Walsh [20] for background on space-time white noise processes and stochastic integration with respect to martingale measures.

Assume that σ : R^d → R^d⊗d and b : R^d → R^d are Lipschitz continuous and with linear growth. Then, by results of [3] or [7] we can construct in some probability space (Ω,F,P) a sequence (Xⁱ)i∈Nof independent copies of the nonlinear processes,

X_tⁱ=X₀ⁱ + Z t

0

Z

R^d

σ(X_sⁱ−y)W_Pⁱ(dy, ds) + Z t

0

Z

R^d

b(X_sⁱ−y)P_s(dy)ds, (6) where the W_Pⁱ are independent space-time R^d-valued white noises defined on [0,∞)×R^d. Each of thed(independent) coordinates ofW_Pⁱ has covariance measure Pt(dy)⊗dt, where P_tis the law ofX_t. The initial conditions (X₀¹, . . . , X₀ⁿ, . . .) are independent and identically distributed with law P₀, and independent of the white noises. The pathwise law of Xⁱ is denoted byP, and it is uniquely determined.

Denote by F_tⁿ the complete right continuous σ-field generated by

{(W_P¹([0, s]×A¹), . . . , W_Pⁿ([0, s]×Aⁿ)) : 0≤s≤t, Aⁱ ∈ B(R^d)}

and (X₀¹, . . . , X₀ⁿ).We also denote by

Predⁿ

the predictable field generated by continuous (F_tⁿ)-adapted processes.

In what follows, we fix a finite time horizonT >0. Under usual Lipschitz assumptions on the coefficients, there is propagation of the moments of the lawP₀, as proved in Gu´erin [7].

Lemma 4.1. If E(|X₀|^k)<∞ for some k≥2, then E sup

t∈[0,T]

|X_t|^k

!

<∞.

The continuity of X and the previous uniform bound imply that t 7→ R

R^d|x|^kP_t(dx) is continuous.

Throughout the sequel, the assumptions of Theorem 1.1 on P₀ and P_t are enforced, in particular, the conditionE(sup_t∈[0,T]|X_t|²)<∞ will hold by the previous lemma.

We shall now present the main idea of the coupling we introduce to prove Theorem 1.1.

Basically, this consists in constructing for eachn,n² Brownian motions in a pathwise way, from the realizations of the n white noises (W_P¹, . . . , W_Pⁿ). The key for that will be to use the optimal transport maps between the marginal Pt of the nonlinear process and the empirical measures of samples of that law. More precisely, write

ν_tⁿ:= 1 n

n

X

i=1

δ_Xⁱ

t

and notice that for each ω ∈Ω, (ν_tⁿ,0 ≤t≤T) is an element of C([0, T],P₂(R^d)). Thus, for each t∈ [0, T], n∈ N and ω, and we can consider the optimal coupling problem with quadratic cost betweenν_tⁿ(ω) andPt,

inf

π<^Pt_νn

t(ω)

Z

R^d×R^d

|x−y|²π(dx, dy)

.

By the assumption on Pt and Theorem 2.5, the following properties hold for each fixed pair(t, ω)∈]0, T]×Ω:

Lemma 4.2. a) There exists a unique π^t,ω,n, such that W₂²(P_t, ν_tⁿ(ω)) =

Z

R^d×R^d

|x−y|²π^t,ω,n(dxdy).

b) There is a Pt(dx)−a.e. unique measurable function T^t,ω,n:R^d→R^d such that π^t,ω,n(dx, dy) =δ_T^t,ω,n_(x)(dy)Pt(dx).

In particular, under Pt(dx) the law of T^t,ω,n(x) isν_tⁿ(ω).

c) We have

W₂²(P_t, ν_tⁿ(ω)) = Z

R²

|x−T^t,ω,n(x)|²P_t(dx).

We would like to construct n² independent Brownian motions by “transporting” the n independent white noises (W_P¹, . . . , W_Pⁿ) through the transport mappings T^s,ω,n(x). As pointed out in the introduction, to do so we must at least be able to define stochastic integrals of functions of the form (t, ω, x) 7→ f(T^t,ω,n(x)), with respect to the white noise processes. The existence of a versionTⁿ(t, ω, x) ofT^t,ω,n(x) having good enough properties, will be established in next section, when we shall prove Theorem 1.2.

Before doing so, we observe that if Theorem 1.2 holds, then the following processes B_t^ik = B_t^ik,n will be well defined from (6).

Proposition 4.3. For each n∈N^∗, define B_t^ik,n(ω) :=√

n Z t

0

Z

R^d

1_{Tn(s,ω,x)=X_s^k(ω)}W_Pⁱ(dx, ds), i, k= 1. . . n (7) Then,(B^ik,n)1≤i,k≤n are n² independent standard Brownian motions in R^d.

These are the right Brownian motions we need to construct (4). The proof of Proposition 4.3 will given in Section 6.

5 Construction of the predictable “transport process”

Our goal now in this section is to show that for each n ∈ N^∗, there exists a process (t, ω, x)7→Tⁿ(t, ω, x) definedP(dω)⊗dt⊗Pt(dx)-almost everywhere, which is measurable with respect toPredⁿ⊗ B(R^d), and such that:

fordt⊗P(dω) almost every (t, ω),